Let $p>3$ be a prime and $(\frac{ .}{ p})$ be the Legendre symbol. For any integer $d$ with $p\nmid d$ and any positive integer $m$, Sun introduced the determinants $$T_m(d,p)=\det\left[(i^2+dj^2)^m\left(\frac{ i^2+dj^2}{ p}\right)\right]_{ 1\leqslant i,j \leqslant (p-1)/2},$$ and $$D_p^{ (m)}= \det\left[(i^2-j^2)^m\left(\frac{ i^2-j^2}{ p}\right)\right]_{ 1\leq i,j\leq (p-1)/2} .$$ In this paper, we obtain some properties of $T_m (d,p)$ and $ \sqrt{ D_p^{ (m)}}$ for some $m$. We also confirm some related conjectures posed by Zhi-Wei Sun.
Categories: math.NT